一、树 & 二叉树
树是由节点和边构成,储存元素的集合。节点分根节点、父节点和子节点的概念。
如图:树深=4; 5是根节点;同样8与3的关系是父子节点关系。
二叉树binary tree,则加了“二叉”(binary),意思是在树中作区分。每个节点至多有两个子(child),left child & right child。二叉树在很多例子中使用,比如二叉树表示算术表达式。
如图:1/8是左节点;2/3是右节点;
二、二叉搜索树 BST
顾名思义,二叉树上又加了个搜索的限制。其要求:每个节点比其左子树元素大,比其右子树元素小。
如图:每个节点比它左子树的任意节点大,而且比它右子树的任意节点小
三、BST Java实现
直接上代码,对应代码分享在Github主页
BinarySearchTree.java
package org.algorithm.tree;/** Copyright [] [Jeff Lee]** Licensed under the Apache License, Version 2.0 (the "License");* you may not use this file except in compliance with the License.* You may obtain a copy of the License at** /licenses/LICENSE-2.0** Unless required by applicable law or agreed to in writing, software* distributed under the License is distributed on an "AS IS" BASIS,* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.* See the License for the specific language governing permissions and* limitations under the License.*//*** 二叉搜索树(BST)实现** Created by bysocket on 16/7/7.*/public class BinarySearchTree {/*** 根节点*/public static TreeNode root;public BinarySearchTree() {this.root = null;}/*** 查找*树深(N) O(lgN)*1. 从root节点开始*2. 比当前节点值小,则找其左节点*3. 比当前节点值大,则找其右节点*4. 与当前节点值相等,查找到返回TRUE*5. 查找完毕未找到,* @param key* @return*/public TreeNode search (int key) {TreeNode current = root;while (current != null&& key != current.value) {if (key < current.value )current = current.left;elsecurrent = current.right;}return current;}/*** 插入*1. 从root节点开始*2. 如果root为空,root为插入值*循环:*3. 如果当前节点值大于插入值,找左节点*4. 如果当前节点值小于插入值,找右节点* @param key* @return*/public TreeNode insert (int key) {// 新增节点TreeNode newNode = new TreeNode(key);// 当前节点TreeNode current = root;// 上个节点TreeNode parent = null;// 如果根节点为空if (current == null) {root = newNode;return newNode;}while (true) {parent = current;if (key < current.value) {current = current.left;if (current == null) {parent.left = newNode;return newNode;}} else {current = current.right;if (current == null) {parent.right = newNode;return newNode;}}}}/*** 删除节点*1.找到删除节点*2.如果删除节点左节点为空 , 右节点也为空;*3.如果删除节点只有一个子节点 右节点 或者 左节点*4.如果删除节点左右子节点都不为空* @param key* @return*/public TreeNode delete (int key) {TreeNode parent = root;TreeNode current = root;boolean isLeftChild = false;// 找到删除节点 及 是否在左子树while (current.value != key) {parent = current;if (current.value > key) {isLeftChild = true;current = current.left;} else {isLeftChild = false;current = current.right;}if (current == null) {return current;}}// 如果删除节点左节点为空 , 右节点也为空if (current.left == null && current.right == null) {if (current == root) {root = null;}// 在左子树if (isLeftChild == true) {parent.left = null;} else {parent.right = null;}}// 如果删除节点只有一个子节点 右节点 或者 左节点else if (current.right == null) {if (current == root) {root = current.left;} else if (isLeftChild) {parent.left = current.left;} else {parent.right = current.left;}}else if (current.left == null) {if (current == root) {root = current.right;} else if (isLeftChild) {parent.left = current.right;} else {parent.right = current.right;}}// 如果删除节点左右子节点都不为空else if (current.left != null && current.right != null) {// 找到删除节点的后继者TreeNode successor = getDeleteSuccessor(current);if (current == root) {root = successor;} else if (isLeftChild) {parent.left = successor;} else {parent.right = successor;}successor.left = current.left;}return current;}/*** 获取删除节点的后继者*删除节点的后继者是在其右节点树种最小的节点* @param deleteNode* @return*/public TreeNode getDeleteSuccessor(TreeNode deleteNode) {// 后继者TreeNode successor = null;TreeNode successorParent = null;TreeNode current = deleteNode.right;while (current != null) {successorParent = successor;successor = current;current = current.left;}// 检查后继者(不可能有左节点树)是否有右节点树// 如果它有右节点树,则替换后继者位置,加到后继者父亲节点的左节点.if (successor != deleteNode.right) {successorParent.left = successor.right;successor.right = deleteNode.right;}return successor;}public void toString(TreeNode root) {if (root != null) {toString(root.left);System.out.print("value = " + root.value + " -> ");toString(root.right);}}}/*** 节点*/class TreeNode {/*** 节点值*/int value;/*** 左节点*/TreeNode left;/*** 右节点*/TreeNode right;public TreeNode(int value) {this.value = value;left = null;right = null;}}
1.节点数据结构
首先定义了节点的数据接口,节点分左节点和右节点及本身节点值。如图
代码如下:
/*** 节点*/class TreeNode {/*** 节点值*/int value;/*** 左节点*/TreeNode left;/*** 右节点*/TreeNode right;public TreeNode(int value) {this.value = value;left = null;right = null;}}
2. 插入
插入,和删除一样会引起二叉搜索树的动态变化。插入相对删处理逻辑相对简单些。如图插入的逻辑:
a. 从root节点开始
b.如果root为空,root为插入值
c.循环:
d.如果当前节点值大于插入值,找左节点
e.如果当前节点值小于插入值,找右节点
代码对应:
/*** 插入*1. 从root节点开始*2. 如果root为空,root为插入值*循环:*3. 如果当前节点值大于插入值,找左节点*4. 如果当前节点值小于插入值,找右节点* @param key* @return*/public TreeNode insert (int key) {// 新增节点TreeNode newNode = new TreeNode(key);// 当前节点TreeNode current = root;// 上个节点TreeNode parent = null;// 如果根节点为空if (current == null) {root = newNode;return newNode;}while (true) {parent = current;if (key < current.value) {current = current.left;if (current == null) {parent.left = newNode;return newNode;}} else {current = current.right;if (current == null) {parent.right = newNode;return newNode;}}}}
3.查找
其算法复杂度 :O(lgN),树深(N)。如图查找逻辑:
a.从root节点开始
b.比当前节点值小,则找其左节点
c.比当前节点值大,则找其右节点
d.与当前节点值相等,查找到返回TRUE
e.查找完毕未找到
代码对应:
/*** 查找*树深(N) O(lgN)*1. 从root节点开始*2. 比当前节点值小,则找其左节点*3. 比当前节点值大,则找其右节点*4. 与当前节点值相等,查找到返回TRUE*5. 查找完毕未找到,* @param key* @return*/public TreeNode search (int key) {TreeNode current = root;while (current != null&& key != current.value) {if (key < current.value )current = current.left;elsecurrent = current.right;}return current;}
4. 删除
首先找到删除节点,其寻找方法:删除节点的后继者是在其右节点树种最小的节点。如图删除对应逻辑:
结果为:
a.找到删除节点
b.如果删除节点左节点为空 , 右节点也为空;
c.如果删除节点只有一个子节点 右节点 或者 左节点
d.如果删除节点左右子节点都不为空
代码对应见上面完整代码。
案例测试代码如下,BinarySearchTreeTest.java
package org.algorithm.tree;/** Copyright [] [Jeff Lee]** Licensed under the Apache License, Version 2.0 (the "License");* you may not use this file except in compliance with the License.* You may obtain a copy of the License at** /licenses/LICENSE-2.0** Unless required by applicable law or agreed to in writing, software* distributed under the License is distributed on an "AS IS" BASIS,* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.* See the License for the specific language governing permissions and* limitations under the License.*//*** 二叉搜索树(BST)测试案例 {@link BinarySearchTree}** Created by bysocket on 16/7/10.*/public class BinarySearchTreeTest {public static void main(String[] args) {BinarySearchTree b = new BinarySearchTree();b.insert(3);b.insert(8);b.insert(1);b.insert(4);b.insert(6);b.insert(2);b.insert(10);b.insert(9);b.insert(20);b.insert(25);// 打印二叉树b.toString(b.root);System.out.println();// 是否存在节点值10TreeNode node01 = b.search(10);System.out.println("是否存在节点值为10 => " + node01.value);// 是否存在节点值11TreeNode node02 = b.search(11);System.out.println("是否存在节点值为11 => " + node02);// 删除节点8TreeNode node03 = b.delete(8);System.out.println("删除节点8 => " + node03.value);b.toString(b.root);}}
运行结果如下:
value = 1 -> value = 2 -> value = 3 -> value = 4 -> value = 6 -> value = 8 -> value = 9 -> value = 10 -> value = 20 -> value = 25 -> 是否存在节点值为10 => 10是否存在节点值为11 => null删除节点8 => 8value = 1 -> value = 2 -> value = 3 -> value = 4 -> value = 6 -> value = 9 -> value = 10 -> value = 20 -> value = 25 ->
四、小结
与偶尔吃一碗“老坛酸菜牛肉面”一样的味道,品味一个算法,比如BST,的时候,总是那种说不出的味道。
树,二叉树的概念
BST算法
相关代码分享在Github主页
转载自并发编程网 – 本文链接地址:Java实现 二叉搜索树算法(BST)